Render just part of a final image from Cycles?
Ctrl + B in the 3D Viewport or the Camera View will allow you to drag while holding  LMB select the area of the image you want to render. If you want to clear the border, press Ctrl + Alt + B.

Shift + B also still works in Camera View but is only there for compatibility and will be replaced entirely by Ctrl + B in future versions.

For "Render->Render Image" (F12 ) to render only part of the image you also need to have Render->Dimensions->Border enabled.